function [ phii, phiix ] = phi ( alpha, beta, i, np, x )

%% PHI evaluates the I-th basis function at the point X.
%
%  Modified:
%
%    03 November 2006
%
%  Author:
%
%    Max Gunzburger
%    Teresa Hodge
%
%    MATLAB translation by John Burkardt
%
%  Parameters:
%
%    Input, real ALPHA(NP).
%    ALPHA(I) contains one of the coefficients of a recurrence
%    relationship that defines the basis functions.
%
%    Input, real BETA(NP).
%    BETA(I) contains one of the coefficients of a recurrence
%    relationship that defines the basis functions.
%
%    Input, integer I, the index of the basis function.
%
%    Input, integer NP.
%    The highest degree polynomial to use.
%
%    Input, real X, the evaluation point.
%
%    Output, real PHII, PHIIX, the value of the basis
%    function and its derivative.
%
  qm1 = 0.0;
  q = 1.0;
  qm1x = 0.0;
  qx = 0.0;

  for j = 1 : i
    qm2 = qm1;
    qm1 = q;
    qm2x = qm1x;
    qm1x = qx;
    t = x - alpha(j);
    q = t * qm1 - beta(j) * qm2;
    qx = qm1 + t * qm1x - beta(j) * qm2x;
  end

  t = 1.0 - x * x;
  phii = t * q;
  phiix = t * qx - 2.0 * x * q;